Solve by using the multiplication principle. $$ \frac{3}{4}=-\frac{5}{12} y $$ The multiplication principle states for any real numbers $$ \mathrm{a}, \mathrm{b} $$, and c , where $$ \mathrm{c}
eq 0 $$, the A. $$ \mathrm{a}=\mathrm{b} $$ is equivalent to $$ \mathrm{a} \cdot \mathrm{c}=\mathrm{b} \cdot \mathrm{c} $$. B. $$ \mathrm{a}=\mathrm{b} $$ is equivalent to $$ \mathrm{b} \cdot \mathrm{a}=\mathrm{c} $$. C. $$ \mathrm{b}=\mathrm{c} $$ is equivalent to $$ \mathrm{a} \cdot \mathrm{c}=\mathrm{b} \cdot \mathrm{c} $$. D. $$ \mathrm{b}=\mathrm{c} $$ is equivalent to $$ \mathrm{b} \cdot \mathrm{c}=\mathrm{a} $$.

Question

Solve by using the multiplication principle. $$ \frac{3}{4}=-\frac{5}{12} y $$ The multiplication principle states for any real numbers $$ \mathrm{a}, \mathrm{b} $$, and c , where $$ \mathrm{c} 
eq 0 $$, the A. $$ \mathrm{a}=\mathrm{b} $$ is equivalent to $$ \mathrm{a} \cdot \mathrm{c}=\mathrm{b} \cdot \mathrm{c} $$. B. $$ \mathrm{a}=\mathrm{b} $$ is equivalent to $$ \mathrm{b} \cdot \mathrm{a}=\mathrm{c} $$. C. $$ \mathrm{b}=\mathrm{c} $$ is equivalent to $$ \mathrm{a} \cdot \mathrm{c}=\mathrm{b} \cdot \mathrm{c} $$. D. $$ \mathrm{b}=\mathrm{c} $$ is equivalent to $$ \mathrm{b} \cdot \mathrm{c}=\mathrm{a} $$.

UpStudy AI · Accepted Answer

To solve the equation $$ \frac{3}{4} = -\frac{5}{12} y $$ using the multiplication principle, we will follow these steps:

1. **Identify the equation**: We have $$ \frac{3}{4} = -\frac{5}{12} y $$.

2. **Apply the multiplication principle**: According to the multiplication principle, we can multiply both sides of the equation by a non-zero constant to maintain equality. In this case, we will multiply both sides by $$ -\frac{12}{5} $$ to isolate $$ y $$.

3. **Perform the multiplication**:
   $$
   y = \frac{3}{4} \cdot \left(-\frac{12}{5}\right)
   $$

4. **Calculate the right side**:
   $$
   y = -\frac{3 \cdot 12}{4 \cdot 5}
   $$

5. **Simplify the expression**:
   - First, calculate $$ 3 \cdot 12 = 36 $$.
   - Then, calculate $$ 4 \cdot 5 = 20 $$.
   - So, we have:
   $$
   y = -\frac{36}{20}
   $$

6. **Reduce the fraction**:
   - The greatest common divisor of 36 and 20 is 4.
   - Dividing both the numerator and the denominator by 4 gives:
   $$
   y = -\frac{9}{5}
   $$

Thus, the solution to the equation $$ \frac{3}{4} = -\frac{5}{12} y $$ is:
$$
y = -\frac{9}{5}
$$

### Regarding the options:
The multiplication principle states that if $$ a = b $$, then $$ a \cdot c = b \cdot c $$ for any $$ c \neq 0 $$. Therefore, the correct answer is:

**A. $$ a = b $$ is equivalent to $$ a \cdot c = b \cdot c $$**.
The solution to the equation $$ \frac{3}{4} = -\frac{5}{12} y $$ is $$ y = -\frac{9}{5} $$. The correct multiplication principle option is A: $$ a = b $$ is equivalent to $$ a \cdot c = b \cdot c $$.

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